# Range of a Matrix

• Mar 25th 2010, 12:58 PM
uli
Range of a Matrix
A Linear Matrix Inequality (LMI) $A(x) > 0$ where $A(x) = \sum_i x_i A_i + B$ and $A_i = A_i^T$ and $A_i , B$ is given. $x$ is a vector in $R^n$.
Then, the linear map $x \rightarrow Cx$ from $R^n$ to $S$, where $S$ is a space of symmetric matrices with prescribed block diagonal structure, is associated such that $A(x) = Cx+B$. The range of this map is $E = range(C)$.

My Question is, how to obtain $E$ ??

I guess, that $C = [A_1,A_2,...]$ is a matrix which columns are the matrices $A_i$.

And $E$ is then the column space of $C$ ??

And when I want to find a matrix which is orthogonal to $E$, is it then a matrix which is orthogonal to $C$ ??

As you might have noticed, I've no idea what to do.

thanks
• Mar 25th 2010, 03:50 PM
tonio
Quote:

Originally Posted by uli
A Linear Matrix Inequality (LMI) $A(x) > 0$ where $A(x) = \sum_i x_i A_i + B$ and $A_i = A_i^T$ and $A_i , B$ is given. $x$ is a vector in $R^n$.

...and thus $Ax$ is a vector in $\mathbb{R}^n$. What does it mean that a vector is greater/less than zero?

Tonio

Then, the linear map $x \rightarrow Cx$ from $R^n$ to $S$, where $S$ is a space of symmetric matrices with prescribed block diagonal structure, is associated such that $A(x) = Cx+B$. The range of this map is $E = range(C)$.

My Question is, how to obtain $E$ ??

I guess, that $C = [A_1,A_2,...]$ is a matrix which columns are the matrices $A_i$.

And $E$ is then the column space of $C$ ??

And when I want to find a matrix which is orthogonal to $E$, is it then a matrix which is orthogonal to $C$ ??

As you might have noticed, I've no idea what to do.

thanks

.
• Mar 26th 2010, 02:27 AM
uli
Quote:

Originally Posted by tonio
...and thus http://www.mathhelpforum.com/math-he...78b3546b-1.gif is a vector in http://www.mathhelpforum.com/math-he...453d1310-1.gif. What does it mean that a vector is greater/less than zero?

Tonio

the vector $x = [x_1 x_2 ...]^T$ contains the decision variables $x_i$, which are scalar, so that the matrix $A(x) = \sum_i x_i A_i + B$ is positive definite. So that it results in an optimization problem which can be solved by using i.e. Semidefinite programming, what I am trying to do.
The matrices $A_i$ and $B$ are symmetric matrices with real elements.

At the time, I stuck with finding $E = range(C)$ and $x \mapsto Cx$ such that $A(x) = Cx + B$.

How to obtain $C$ and remains the vector $x$ a vector or becomes $x$ a matrix after mapping since $Cx$ must be of same dimension as $B$ because of adding this two matrices ??

and then, how to get $E$ ??